What is the integral of (sqrt(x^2-1))/(2x^2) ?

The integral of (sqrt(x^2-1))/(2x^2) is 1/2 (-(sqrt(x^2-1))/x+ln|sqrt(x^2-1)+x|)+C

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integral of (sqrt(x^2-1))/(2x^2)

\int \frac{x ^{2} - 1}{2 x ^{2}} d x

\frac{1}{2} (- \frac{x ^{2} - 1}{x} + ln x^{2} - 1 + x) + C

Solution steps

\int \frac{x ^{2} - 1}{2 x ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{x ^{2} - 1}{x ^{2}} d x

Apply Integration By Parts

: - \frac{x ^{2} - 1}{x} - \int - \frac{1}{x ^{2} - 1} d x

= \frac{1}{2} (- \frac{x ^{2} - 1}{x} - \int - \frac{1}{x ^{2} - 1} d x)

\int - \frac{1}{x ^{2} - 1} d x = - ln x^{2} - 1 + x

= \frac{1}{2} (- \frac{x ^{2} - 1}{x} - (- ln x^{2} - 1 + x))

Simplify

= \frac{1}{2} (- \frac{x ^{2} - 1}{x} + ln x^{2} - 1 + x)

Add a constant to the solution

= \frac{1}{2} (- \frac{x ^{2} - 1}{x} + ln x^{2} - 1 + x) + C

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What is the integral of (sqrt(x^2-1))/(2x^2) ?
The integral of (sqrt(x^2-1))/(2x^2) is 1/2 (-(sqrt(x^2-1))/x+ln|sqrt(x^2-1)+x|)+C